Article pubs.acs.org/Langmuir
Viscosity-Dependent Janus Particle Chain Dynamics Bin Ren† and Ilona Kretzschmar*,†,‡ †
Department of Chemistry, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, United States ‡ Department of Chemical Engineering, City College of New York, City University of New York, 140th ST. and Convent Ave., New York, New York 10031, United States S Supporting Information *
ABSTRACT: Iron oxide (Fe3O4) Janus particles assemble into staggered chains parallel to the field lines in an ac electric field. Subsequent application of an external magnetic field leads to contraction of the staggered chains into double chains. The relation between the viscosity of the surrounding solution and the contraction rate of the iron oxide Janus particle chains is studied. Further, the influence of particle size and chain length (i.e., number of particles in chain) on the contraction rate is investigated. The base material for the Janus structure is silica (SiO2) with particle sizes of 1, 2, and 4 μm, and the cap material is Fe3O4. Addition of increasing amounts of glycerol to the aqueous system reveals that the contraction dynamics strongly correlate with the viscosity of the solution. The average chain contraction rate for each particle size can be fitted in the low viscosity range from 1 to 30 mPa·s with a power function of the form A/μ0.9 − B/μ, in which the coefficients A and B are particle size, electric field, and magnetic-field-dependent constants. Using this function, the viscosity of an unknown solution can be determined, thereby pointing to the potential application of these Janus particle chain assemblies as in situ microviscometers.
1. INTRODUCTION Measuring the viscosity of chemical and biological fluids is a key issue in many industries ranging from chemical and manufacturing to pharmaceutical and food processing industries1 because the ability of the fluid microstructure to rearrange under an imposed flow determines the macroscopic rheological response of the fluid. For example, in the area of quality control, the viscosity is measured to ensure batch-to-batch consistency of raw materials. Further, in the design of pumping and piping systems the viscosity of the target liquid is required since a high-viscosity liquid requires more power to pump than a low-viscosity one. Viscosity measurements are also regularly employed as a material characterization method because the flow behavior is responsive to material properties such as molecular weight and arrangements. The viscosity of a Newtonian fluid can be measured using a viscometer at a specific flow condition, whereas non-Newtonian fluids require a rheometer because their viscosities depend on the flow conditions. There are many types of viscometers that have been developed to measure the viscosity of a Newtonian fluid such as the falling sphere method2,3 as well as capillary,4−8 rotational/sliding,9,10 vibrational,11 and microfluidic viscometers.12,13 The falling sphere method obtains the viscosity based on Stokes’ law, which states that when a sphere of known size falling in a liquid reaches its terminal velocity, the fractional force is balanced with the gravitational force.14 In contrast, © 2013 American Chemical Society
capillary viscometers measure the time it takes a known volume of liquid to pass through a capillary of known diameter, which is proportional to the kinematic viscosity. More modern capillary and microfluidic viscometers measure the pressure difference between the two ends of a capillary tube/channel with the liquid flowing at a known rate.15 In this case, the viscosity is calculated directly from Poiseuille’s law. The rotational viscometer measures the needed force (shear stress) applied to a rotational part to keep a constant shear velocity. The viscosity of the test fluid is calculated using the shear velocity, surface area of the rotational part, and the measured force. The vibrational viscometer measures the damping of an oscillating electromechanical resonator immersed in a fluid. The higher the viscosity, the larger the damping imposed on the resonator. These various viscometry methods have in common that they contain the liquid they are measuring. As a result, the sample volumes required for measurements range from a few nanoliters to several milliliters.8,16 The viscosity range accessible with these viscometers is 1 × 10−3−1 × 102 Pa·s.7,13,15,17 In addition, traditional viscometers often have complicated setups and are expensive. Therefore, a low-cost and simple viscometer setup that requires only small sample volumes is highly desirable. Received: July 9, 2013 Revised: November 11, 2013 Published: November 12, 2013 14779
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microviscometer based on the behavior of such Fe3O4-capped Janus particle chains in solutions of varying viscosity, which could be used to measure (i) viscosities of organic liquids for which only very limited sample volumes are available or (ii) in situ viscosities of closed systems such as cells or lab-on-a-chip devices. The low-viscosity response (μ = 1−30 mPa·s) of Janus particle chains is calibrated using different ratios of glycerol (μG = 1.450 Pa·s) and water (μw = 1 × 10−3 Pa·s),24 hereafter referred to as the G:W ratio. We show that the average contraction rate of the Janus particle chains decreases with increasing viscosity of the surrounding medium and also scales with particle size.
The asymmetric nature of Janus particles inspired Kopelman et al.18,19 to apply magnetic modulated optical nanoprobes (MagMOONs) and their viscosity-dependent nonlinear rotational behavior in rotational nanoviscometers. Such a nanoviscometer measures kinematic viscosities in the range from 5 × 10−5−3 × 10−4 m2/s in the presence of a rotational magnetic field.20 Kopelman et al.21,22 developed this concept further in their torque-based asynchronous magnetic bead rotation (AMBR) sensor, which enables the measurement of the nanoscale growth dynamics of individual bacterial cells. The limitation of these MagMOON and AMBR particle-based techniques is the need for a clear solution to observe the particle cap orientation and a complex rotational magnetic field setup. In recent work,23 we identified a semiconducting and ferrimagnetic iron oxide (Fe3O4) capping material for Janus particles that yields staggered and double chains in external ac electric and magnetic fields, respectively. The two chain configurations differ in length by ∼40%, yielding a visible chain length reduction during the configurational change (Figure 1). Here, we present the idea of an in situ
2. EXPERIMENTAL SECTION 2.1. Materials. 1.17 ± 0.05, 2.06 ± 0.05, and 4.06 ± 0.21 μm monodispersed silica (SiO2) powders by AngstromSphere are purchased from Fiber Optic Center, Inc. The as-bought SiO2 powder is dispersed in deionized (DI) water (R = 18.2 MΩ) at a concentration of 30 wt % for monolayer assembly on microscope glass slides (Fisher Scientific, Inc.). 1/8 in. × 1/8 in. iron (Fe) evaporation pellets (99.95%) from Kurt J. Lesker Company and three-strand tungsten wire baskets (Ted Pella, Inc.) are used for the iron deposition in a 3:1 Ar/O2 mixture (Airgas Inc.). Glycerol (spectrophotometric grade, 99.5+%) from Acros Inc. is used to adjust the viscosity of the aqueous particle solution. 2.2. Rheological Measurements. The viscosity of glycerol and glycerol:DI-water solutions is measured with an AR 2000 EX rheometer from TA Instruments Inc. at 20 °C. Pure water (1.5 mL) and four G:W ratios are used: 1:4 (0.3/1.2 mL), 1:2 (0.5/1 mL), 1:1 (0.75/0.75 mL), and 2:1 (1/0.5 mL). Their viscosities are measured as μ(0:1) = 1 × 10−3 Pa·s, μ(1:4) = (2.6 ± 0.5) × 10−3 Pa·s, μ(1:2) = (4.3 ± 0.6) × 10−3 Pa·s, μ(1:1) = (9.7 ± 1.0) × 10−3 Pa·s, and μ(2:1) = (29.6 ± 2.4) × 10−3 Pa·s. As expected, the viscosity of the glycerol:water mixture increases with the amount of glycerol added. The viscosity curve at 20 °C as a function of the G:W ratio is provided in the Supporting Information (Figure S1). 2.3. Janus Particle Fabrication and Chain Dynamics Measurement. Janus particles are prepared as reported previously.25 In brief, SiO2 particle monolayers are made using an NE-1000 programmable syringe pump (New Era Pump Systems, Inc.). 50 nm of iron oxide (Fe3O4) is deposited on top of the monolayer using a
Figure 1. Schematic of staggered Janus particle assembly in ac electric field, E (step 1), and subsequent contraction to double-chain configuration upon application of a parallel, static magnetic field, B (step 2).
Figure 2. Snapshots of a 4 μm Fe3O4 Janus particle chain contracting in parallel applied ac electric and static magnetic fields. The white dot indicates the position of the chain center of mass at application of the static magnetic field (t0). The time interval, Δt, between two images is 0.1 s. Scale bar is 10 μm. L1 and L2 indicate the chain length before and after contraction, respectively. 14780
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Figure 3. Average particle−particle center distance, r, in the direction parallel to the external magnetic field as a function of time, t, for chains formed from 1, 2, and 4 μm Fe3O4 Janus particles undergoing the configurational change from staggered to double chain in (A) pure water (μw = 1 × 10−3 Pa·s) and G:W ratios of (B) 1:4 (μ1:4 = (2.6 ± 0.5) × 10−3 Pa·s), (C) 1:2 (μ1:2 = (4.3 ± 0.6) × 10−3 Pa·s), and (D) 1:1 (μ1:1 = (9.7 ± 1.0) × 10−3 Pa· s). The static magnetic field is applied at t0 = 0 s. The top (blue), middle (red), and bottom (black symbols) data series correspond to 4, 2, and 1 μm Janus particle chains, respectively. Different symbols within the same data series (color) indicate chains with varying number of particles of the same particle size. The solid, dashed, and dotted lines indicate r values of 4.06, 2.06, and 1.17 μm, respectively. benchtop metal evaporator system (Cressington 308 R, Ted Pella, Inc.) with a 3:1 Ar:O2 background gas mixture at a pressure of 1 × 10−3 mbar and a deposition rate in the range from 0.22 to 0.33 nm/s yielding Fe3O4. Subsequently, the Fe3O4 Janus particles are redispersed into the solution medium and exposed to an ac electric field followed by parallel application of an external magnetic field. The setup is described in detail elsewhere,23 and a schematic drawing and additional experimental details are provided in the Supporting Information (Figure S2). For each experiment, 40 μL of the 0.07% w/v iron oxide Janus particle dispersion is placed into a cylindrical cell (D = 1 cm, h = 80 μm) and covered with a microscope cover slide to prevent evaporation. The Janus particles randomly disperse in the cell, and the solution is allowed to equilibrate for ∼1 min prior to application of the ac electric field (112.6 V, 75 kHz) for 5 min to induce staggered chain formation within the assembly cell. Upon formation, chains settle in a plane located about ∼1−2 particle diameters above the bottom wall of the cell and retain their Brownian fluctuations. Then the 0.008 T U-shaped permanent magnet is placed around the sample to induce a constant external magnetic field while the ac electric field is kept in place. Videos are taken with a UI-2240-C camera (Imaging Development Systems GmbH) mounted on an Olympus BX51 optical microscope to capture the configurational change from staggered to double chain. The videos are taken at 10 frames/s, and all effective frames are analyzed using the procedure described in the Supporting Information. The contraction time, tc, is calculated in seconds by dividing the number of effective frames by the frame rate. Average contraction rates, kc,avg, reported are averages of at least eight chains per particle size and viscosity. In the case of the 4 μm Fe3O4-capped SiO2 particles in DI water, a high-speed camera (Photron, Fastcam SA3, Model 120 K-M3 LCA) is used to facilitate a more detailed chain contraction process analysis.
3. RESULTS The chain contraction process has been studied for Fe3O4 Janus particles with diameters of 1, 2, and 4 μm in five viscosities. Here, we present (i) the Fe3O4 Janus particle chain contraction mechanism, (ii) the chain contraction rate as a function of time, particle size, number of particles in the chain, and viscosity, and (iii) a detailed analysis of the Janus particle chain contraction rate using high-speed camera data. Figure 2 shows a series of snapshots of a typical staggered 4 μm Fe3O4-capped SiO2 Janus particle chain contracting into a double chain in DI water in the presence of the external ac electric and magnetic fields (Figure 1, step 2). The chain with length, L1, forms upon application of the ac electric field (not shown) and addition of the magnetic field occurs at t0 = 0 s. Both fields are aligned such that their field lines orient from top to bottom of the image. The time interval between any two images is 0.1 s. The blurriness at the end of the chains in the images at t = 0.5 and 0.6 s is due to the fast motion of the particles during chain collapse. The images suggest that the chain contracts from the two ends toward the center of mass of the chain (original position of center of mass is indicated by the white dot), which does not move during the contraction of the chain to L2. During contraction, the Fe3O4 caps that face toward the center of the chain connect to each other on both sides of the chain to form a wider continuous magnetic strip along the center of the chain (Figure 1, step 2). In order to determine the effect of viscosity and particle size on the chain contraction process, 1, 2, and 4 μm Fe3O4-capped 14781
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Figure 4 summarizes the high-speed contraction analysis for one 4 μm Fe3O4-capped SiO2 Janus particle chain in water
SiO2 Janus particles are dispersed into solutions with known viscosities. Figure 3 shows average particle−particle center distance, r (in the direction parallel to the external magnetic field), versus time, t, curves for the three particle sizes in (A) pure water (μw = 1 × 10−3 Pa·s) and G:W mixtures of (B) 1:4 (μ1:4 = (2.6 ± 0.5) × 10−3 Pa·s), (C) 1:2 (μ1:2 = (4.3 ± 0.6) × 10−3 Pa·s), and (D) 1:1 (μ1:1 = (9.7 ± 1.0) × 10−3 Pa·s). The data for the 2:1 (μ2:1 = (29.6 ± 2.4) × 10−3 Pa·s) system is given in the Supporting Information (Figure S3). Top (blue), middle (red), and bottom (black symbols) data series in each graph correspond to 4, 2, and 1 μm Fe3O4-capped SiO2 Janus particles, respectively. Each curve plotted in Figure 3 starts at t0 = 0 s, i.e., the time when the magnetic field is applied and the chain has not contracted yet. The average particle−particle center distance, r, for each chain is calculated using r = 2(L − d)/(n − 1), where L is the total chain length measured by ImageJ as illustrated in Figure 2, d is the average diameter of the particle as provided by AngstromSphere (see Experimental Section), (L − d) is the particle center-to-center distance of the two outmost particles in a chain, and n is number of particles in the chain. Normalizing (L − d) by (n − 1) and multiplying by 2 results in average r values that are independent of chain length and thus collapse all curves for a particular particle size onto a single master curve. The r values for the collapsed chains, i.e., at longer times, line up well with the average diameter values of 1.17, 2.06, and 4.06 μm (dotted, dashed, and solid lines, respectively) since the particle−particle center distance is equivalent to the diameter in the close-packed double chain. The slight translation shift of the curves for a specific particle size (most prominent for 1 μm particles) is a reflection of the size distribution of the silica particles, the batch-to-batch cap material variation, and small variations in the chain structure as well as the accuracy with which the chain length can be determined using optical microscopy. The average particle chain contraction rates, kc,avg, shown in Table 1 are obtained from three batches of samples that have
Figure 4. Chain contraction process monitored at 2000 frames/s for a chain composed of seven 4 μm Fe3O4-capped SiO2 Janus particles contracting in pure water (G:W = 0:1). (A) Total chain length, L, as a function of time, t. Insets show chain before and after contraction. (B) Particle−particle center distance, rP−P, in the direction parallel to the external magnetic field between two adjacent particles positioned on the same side of the chain as a function of time, t. The inset shows the chain prior to application of the magnetic field with particles labeled consecutively. Symbols indicate particle pairs: blue squares, P1 and P3; red circles, P3 and P5; orange up-triangles, P5 and P7; black downtriangles, P2 and P4; and yellow diamond, P4 and P6.
Table 1. Average Chain Contraction Rates, kc,avg, in μm/s for 1, 2, and 4 μm Fe3O4-Capped SiO2 Janus Particle Chains in Various G:W Ratios G:W ratio
water
1:4
1:2
1:1
2:1
1 μm 2 μm 4 μm
3.4 ± 0.6 6.2 ± 0.5 7.6 ± 0.7
3.0 ± 0.6 5.0 ± 0.4 6.7 ± 0.5
2.0 ± 0.6 3.4 ± 0.7 4.7 ± 0.7
1.3 ± 0.4 2.0 ± 0.5 3.2 ± 0.8
0.9 ± 0.2 1.3 ± 0.6 2.0 ± 0.9
(G:W = 0:1). The total chain length, L, and the change in particle−particle center distance, rP−P, are given as a function of time, t, upon application of the magnetic field in Figures 4A and 4B, respectively. There are seven particles in the chain (see inset Figure 4A), the initial chain length is Li = 23.8 μm, and the final chain length is Lf = 15.3 μm, resulting in a change in chain length of ΔL = 8.5 μm. t0 = 0 s represents the time at which the magnetic field is added. The chain contraction process is recorded in 2147 effective frames at 2000 frames/s, which results in a contraction time of tc = 1.1 s and a contraction rate of kc = 7.7 μm/s. Each colored curve in Figure 4B represents rP−P as a function of time for a set of two particles that are situated on the same side of the chain. Particles are numbered consecutively starting at the top of the chain as indicated in the inset of Figure 4B. Five particle pairs are analyzed P1−P3, P3−P5, P5−P7, P2−P4, and P4−P6. The contraction process of the P3−P5 particle pair begins 0.51 s after the addition of the magnetic field, and P1−P3, P5−P7, P2−P4, and P4−P6 start to contract 154, 98, 250, and 7 ms after the P3−P5 particle pair, indicating that gaps close in a random sequence along the chain. Differences in initial rP−P values are attributed to the slight nonuniformity of the chain
been characterized for chain assembly and contraction. The Janus particles for all three batches are fabricated under identical conditions. kc,avg is calculated by averaging the contraction rate of at least eight measurements at a particular viscosity and a particular particle size (see Table S1 in Supporting Information). The error is given as one standard deviation. Note that the chains analyzed in Figure 3 have varying lengths indicated by different symbols. Plotting kc of a specific chain versus the number of particles in that chain at various viscosities shows no distinct correlation at G:W ratios of 0:1, 1:4, 1:2, and 1:1, whereas a weak negative correlation of kc with the number of particles is observed at a G:W ratio of 2:1 (see Supporting Information, Figure S4). As can be seen in Figures 2 and 3, the process of chain collapse occurs within 1 s. Thus, in order to analyze the collapse process in more detail, a high-speed camera (2000 frames/s) is used. 14782
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structure and size distribution of the particles. Effective frames used for analysis start at a Δr > 1.0% and end when Δr < 1.0%. Table 2 gives the contraction rates, kc(P−P), for each particle pair and the geometric parameters obtained from the image analysis. Table 2. Contraction Rates of Particle Pairs, kc(P−P), in a 4 μm Fe3O4-Capped SiO2 Janus Particle Chain in the Presence of the AC Electric Field upon Addition of an External Magnetic Field of 0.008 T
P1−P3 P3−P5 P5−P7 P2−P4 P4−P6
effective frames
tc (s)
rinitial (μm)
rfinal (μm)
ΔrP−P (μm)
kc(P−P) (μm/s)
1122 1675 1033 801 1444
0.56 0.84 0.25 0.40 0.72
6.68 6.42 6.39 6.64 6.21
3.71 3.67 3.85 3.93 3.81
2.98 2.75 2.54 2.70 2.40
5.30 3.28 4.91 6.75 3.33
The fact that the five curves in Figure 4B have nearly identical shapes and overlap allows the conclusion that the five gaps close with a similar rate and within a narrow time interval (tc = 0.6 ± 0.2 s), indicating that the contraction is caused by instantaneous magnetization of and resulting magnetic dipole− dipole interactions between the Fe3O4 caps. The average contraction rate of all five pairs of particles is calculated as kc(P−P)avg = 4.7 ± 1.5 μm/s.
4. DISCUSSION The Fe3O4 cap endows the Janus particles with a semiconducting property resulting in the assembly of particles in a staggered chain configuration upon exposure to an external ac electric field owing to the electric dipole−dipole interactions between the caps (Figure 1, step 1).26 Upon application of a static magnetic field, the ferrimagentic nature of the Fe3O4 cap material (Ms = 445 emu/cm2)23 leads to an instantaneous magnetization of the caps along the long axis of the cap (i.e., along the cap base) and subsequent contraction of the staggered chain configuration into a double-chain configuration due to magnetic dipole−dipole interactions (Figure 1, step 2). The cause for the change from staggered to double chain configuration is not well understood; however, preliminary simulation results indicate that the configuration change is a result of a shifted magnetic dipole in the Janus cap.27 Monitoring the center of mass position of the chains during the contraction process (Figure 2) reveals that the chains contract toward their center of mass, while the center of mass does not move. In addition, high-resolution imaging reveals that particle pairs close at a comparable rate in pure water (Figure 4). These observations have two implications: (i) the outmost particles of the chain will experience the highest velocity (vP) and largest drag force (Fdrag ∼ μRvP) during contraction, i.e., the chain contraction rate, kc, should be viscosity (μ) and particle-size dependent (d = 2R), and (ii) kc should correlate with chain length (L), i.e., a longer chain requires the outmost particle to move faster to cover a longer distance (ΔL) in the same amount of time (tc) assuming that the gaps between particle pairs close simultaneously along the chain. Figure 5 displays the experimental average contraction rates, kc,avg, obtained for (A) 4 μm, (B) 2 μm, and (C) 1 μm Janus particles at five viscosities (Figure 3 and Figure S3). For each particle size, kc,avg decreases exponentially with increasing viscosity and more linearly with decreasing particle size, in
Figure 5. Plot of average contraction rate, kc,avg, as a function of solution viscosity, μ, for (A) 4 μm (blue squares), (B) 2 μm (red circles), and (C) 1 μm (black triangles) Fe3O4-capped SiO2 Janus particle chains. The solid lines are least-squares fits obtained by adjusting the coefficients A and B in eq 7 (see text for details).
good agreement with implication (i) mentioned above. After a pronounced drop in rate of more than 40% in the (1−5) × 10−3 Pa·s region, the contraction rate decreases more slowly (∼5%) in the (5−10) × 10−3 Pa·s range. Surprisingly, at viscosities above 10 × 10−3 Pa·s, the contraction rates are nonzero and nearly constant, i.e., Δkc,avg < 0.1%. The error bars in Figure 5 represent one standard deviation. Four factors contribute to the uncertainty of the kc,avg values: (i) the batchto-batch variation in the cap material as indicated by the range of resistances (2.7 ± 0.7 to 110 ± 50 kΩ) reported in ref 25, which affect both the electric field and the magnetic field interactions, (ii) the variation in particle size, which leads to a variation in the drag force, (iii) slight variations in chain structure resulting from defects in the caps, and (iv) the accuracy of chain length measurement due to the limited optical resolution. Some of those factors can be further minimized by, for example, choosing a different capping material, using larger magnetic and/or electric fields, or employing a less packed monolayer to avoid cap defects. In order to apply this structural transition as a tool for viscosity measurements, kc,avg has to be fitted with a well-defined relation between kc and the viscosity, μ, of the surrounding medium, 14783
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Rearranging eq 5 for kc and replacing εr as shown in eq 3b yields eq 6 p1 p2 μ m1m2 1 1 kc = − 02 4 2 4 0.9 955π Rε0r μ 2π Rr μ (6)
which can be found by analysis of the forces acting on the particles in the chain. In a first approximation, three forces are at work when the chain undergoes the configurational change: (i) the magnetic dipole−dipole interaction, Fmag, which results from magnetization of the Fe3O4 caps and initiates the collapse, (ii) the electric dipole−dipole interaction, Felectric, which opposes the collapse due to the applied ac electric field, and (iii) the drag force, Fdrag, which opposes the particle movement due to the surrounding liquid and is largest at the two ends of the particle chain. When the velocity of the particles reaches a constant rate during the contraction, the sum of the magnetic and electric dipole−dipole interaction forces and the drag force has to be zero, leading to the force balance given in eq 1, when inertial forces are neglected (note Fdrag ≫ Finertial by 4 orders of magnitude in the experiments reported here):28,29 Fmag − Felectric = −Fdrag
which can be further simplified into eq 7 by combination of all electric and magnetic constants into coefficients A and B, respectively: A B kc = 0.9 − μ μ (7) Equation 7 shows that the chain contraction rate has an exponentially decreasing relationship with the viscosity of the surrounding medium, in good agreement with the experimental findings (Figure 5). In addition, kc is a function of the radius of the particle and depends on the magnetic and electric properties of the cap material. Fitting of eq 7 to the experimental data depicted in Figure 5 yields values for coefficients A and B that are summarized in Table 3. A least-
(1)
The magnetic dipole−dipole interaction force can be expressed by eq 2, in which m1 and m2 are the magnetic dipole moments of two interacting Janus particles, μ0 = 4π × 10−7 N/A2 is the vacuum permeability, and r is the distance between two particle centers. The magnitude of the magnetic dipole moment of the Fe3O4 cap is determined by the size of the particle and the thickness of the Fe3O4 cap. Since the magnetic field is constant (B = 0.008 T), r is the only variable in eq 2.30 2
6μ0 m1 m2
Fmag =
Table 3. Values and Ranges for Coefficients A and B Obtained from Least Squares Fitting (LSF) Procedure and Matlab Analysis (MA), Respectively, of kc,avg vs μ Data Depicted in Figure 5 1 μm
2
4πr 4
A = p1p2/ 955π2Rε0r4 B = μ0m1m2/ 2π2r4
(2)
The electric dipole−dipole interaction force can be calculated by eq 3a, in which p1 and p2 are the electric dipole moments of two interacting particles, ε0 = 8.854 × 10−12 F/m is the vacuum permittivity, and εr is the complex dielectric constant.31 pp 1 Felectric = 1 2 4 4πε0r εr (3a)
εr = 79.6μ−0.1
Since glycerol is used to adjust the viscosity of the aqueous solution, the complex dielectric constant εr varies with the ratio of glycerol (εglycerol = 42.5) to water (εwater = 80.7).32 The specific relation between μ and εr of the glycerol:water system is given in eq 3b, and its derivation is presented in the Supporting Information (Figure S5). The drag force results from the friction experienced by the particles when moving through the liquid during contraction toward the chain center, and is expressed in eq 4a, which is also known as Stokes’ law.33 (4a)
kc = 2vp
(4b)
In eq 4a, μ is the viscosity of the fluid, R is an effective particle radius, and vp is the velocity of the outmost particle, which is related to the measured contraction rate, kc, by eq 4b since both particles at the two ends of the chain contribute to the chain length change ΔL during tc, which is used to calculate kc = ΔL/ tc. Insertion of eqs 2, 3, and 4a into eq 1 yields eq 5: 6μ0 m12m2 2 4πr
4
−
p1 p2 1 = −3πμRkc 4πε0r 4 εr
4 μm
MA
LSF
MA
LSF
MA
40
39−51
60
58−71
93
83−102
36
35−48
54
51−65
86
75−95
squares fitting (LSF) procedure is used to obtain the best fit, and a Matlab analysis (MA) routine is used to determine the range for coefficients A and B (see Supporting Information). Interestingly, coefficients A and B are very comparable in size with A being always slightly larger than B. From Table 3, it is apparent that coefficients A and B strongly depend on particle size. This observation is in good agreement with the fact that these coefficients are a collection of electric and magnetic properties, respectively, which depend on the Fe3O4 cap size. The Fe3O4 cap is proportional to the square of the particle radius. The cap volume of the 4 μm Fe3O4 Janus particles is 4 and 15 times larger than that of a 2 and 1 μm Janus particle, respectively. Thus, the 4 μm Janus particles experience the strongest magnetic dipole−dipole interaction when the magnetic field is applied, while the 2 μm Janus particles have less magnetic interaction with the 1 μm Janus particle−particle interaction being the weakest. The same can be said for the electric properties, coefficient B. Implication (ii) predicted the dependence of the kc on chain length. As shown in Figure S4, no distinct linear correlation between chain length and kc is observed for the kc values at G:W ratios of 0:1, 1:4, 1:2, and 1:1; however, at the highest viscosity a weak negative linear correlation is observed. The rationale for this finding is that while particle−particle gaps within one chain close at a similar rate, i.e., the magnetic dipole−dipole interaction is not affected by the changing glycerol:water ratio, the interval between particle−particle gap closures increases with increasing viscosity. Figure 6 depicts the number of gaps closed for a representative set of 4 μm Fe3O4 Janus particle chains at the five viscosities as a function of time. Note t = 0 s is the time
(3b)
Fdrag = −6πμRvp = −3πμRkc
2 μm
LSF
(5) 14784
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carbons) have viscosities below 30 × 10−3 Pa·s.35 In addition, a more magnetic cap material or a stronger magnetic field could be used to increase the accessible viscosity range. The reversibility of the chain contraction is the subject of ongoing research. The advantages of the microviscometer are that (i) it requires only picoliter volumes of fluid and (ii) the setup is straightforward; i.e., Fe3O4 Janus particles are added to the target fluid, followed by external application of first an electric and then a second magnetic field to assemble the staggered chain and observe its contraction.
5. CONCLUSION The contraction of staggered chains formed in an external ac electric field to double chains upon application of a parallel magnetic field has been analyzed for 1, 2, and 4 μm Fe3O4capped SiO2 Janus particles. The chain contraction processes is recorded at viscosities of 1 × 10−3, (2.6 ± 0.5) × 10−3, (4.3 ± 0.6) × 10−3, (9.7 ± 1.0) × 10−3, and (29.6 ± 2.4) × 10−3 Pa·s, yielding average chain contraction rates, kc,avg, that decreases with increasing viscosity. We find that kc,avg is also particle size dependent; i.e., it decreases with decreasing particle size and shows a weak dependence on chain length at the highest viscosity. Calibration curves for 1, 2, and 4 μm Fe3O4 Janus particle chain contraction rates vs viscosity are obtained that can be fitted to a functional of the form kc = A/μ0.9 − B/μ derived from a simple, first-order force balance of electric, magnetic, and drag forces. According to this calibration curve, the corresponding viscosity at a measured chain contraction rate can be determined, illustrating the potential use of Fe3O4 Janus particle chains as in situ microviscometers.
Figure 6. Number of closed particle pairs along a chain comprised of 4 μm Fe3O4 Janus particles vs time elapsed since addition of magnetic field. Colors indicate various G:W ratios (black empty squares, 0:1; red solid circles, 1:4; green empty triangles, 1:2; blue solid triangle, 1:1; and purple solid square, 2:1).
when the magnetic field is added. With increasing viscosity, the slope of the lines increases from 0.05 to 0.62, indicating longer intervals between particle−particle gap closures. The intercept with the time axis also increases with increasing viscosity from 0.43 to 1.52, indicating that the time interval between addition of the magnetic field and closure of the first gap increases in good agreement with the more delayed onset of contraction observed with increased viscosity (Figure 3 and Figure S3). A simple force balance model has been employed, which likely does not account for all forces acting within the system. We have limited the model to include forces due to the external electric and magnetic fields and assumed them to be balanced with the drag force acting on the outermost particles in the chain. In addition, we are only considering electric and magnetic dipole interactions parallel to the external fields along the particle−particle center distance. Further, only the dipole−dipole interaction between two-particle pairs (1−3, 3− 5, 5−7, etc., as shown in Figure 4) is considered. Additional interactions beyond those particle pairs are ignored due to the short ranged interaction of dipole−dipole forces (r−4 dependence). As mentioned in the Experimental Section, the particle chains are formed near the bottom of the cell and thus are likely to have some interaction with the cell wall. Both the particles and the cell wall are negatively charged in aqueous solution. However, the addition of glycerol changes the dielectric constant of the medium, thereby reducing the Debye length (by ∼20%). A smaller Debye length will bring the chains closer to the wall and may result in stickiness of the wall, potentially explaining the delay in chain contraction (Figures 3 and 6). However, at the same time, lubrication forces34 may arise as the chains move closer to the wall, leading to the underestimation of kc,avg values at the highest viscosity (Figure 5). Overall, excellent agreement between the kc,avg values and the model is found for all three particle sizes, justifying the simple force balance model as a first approximation. The measurements presented and analyzed here use 40 μL of a 0.07% w/v Janus particle suspension. However, only 8−17 of the formed chains containing 4−14 Janus particles were analyzed to obtain kc,avg. The maximum number of 238 particles can be easily distributed into a volume of a few picoliters. Two limitations of the proposed in situ microviscometer at this time are its upper viscosity limit of 30 × 10−3 Pa·s and its irreversible chain contraction. Fortunately, many aqueous suspensions and organic compounds (fewer than 18
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ASSOCIATED CONTENT
* Supporting Information S
Additional information on viscosity measurements, measurement setup, image analysis, chain contraction rate data, chain length correlation data, and derivation of eq 7. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (I.K.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This project was supported by the National Science Foundation under CAREER Award NSF-CBET 0644789. The authors acknowledge Prof. M. Gherasimova for helpful discussions. B.R. acknowledges Dr. Rui Zhang for help with Matlab coding.
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REFERENCES
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